Localization Principle in Variational Relaxation

• Localization Principle is a structural condition on Borel-measurable integrands that ensures local recovery of gradient information and equi-integrability.
• It enables the replacement of arbitrary bounded-energy Sobolev sequences with competitors having affine boundary structure, preserving local behavior.
• The principle underpins integral representation formulas for Lᵖ-relaxation, accommodating both polynomial and exponential growth integrands.

The localization principle in the sense of Mandallena is a structural condition on a Borel-measurable, possibly extended-valued integrand $L:\mathbb{M}\to[0,\infty]$ that mediates between local approximation, Young-measure analysis, lower semicontinuity, and relaxation for multiple integrals on $W^{1,p}(\Omega;\mathbb{R}^m)$ with $p\in]1,\infty[$. In this framework, the principle is formulated as the condition $(C_{p,q})$, and it is used to derive both a general sequential lower-semicontinuity theorem and integral representation formulas for the $L^p$-relaxation of nonconvex functionals, including cases with exponential growth (Mandallena, 2011).

1. Formal setting and definition

Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain, let $m,n\ge 1$, let $\mathbb{M}=\mathbb{R}^{m\times n}$, let $1<p<\infty$, let $1\le q\le\infty$, and let $W^{1,p}(\Omega;\mathbb{R}^m)$0 be Borel-measurable and $W^{1,p}(\Omega;\mathbb{R}^m)$1-coercive, meaning that there exists $W^{1,p}(\Omega;\mathbb{R}^m)$2 such that

$W^{1,p}(\Omega;\mathbb{R}^m)$3

Set

$W^{1,p}(\Omega;\mathbb{R}^m)$4

The localization principle $W^{1,p}(\Omega;\mathbb{R}^m)$5 is the following statement. For every $W^{1,p}(\Omega;\mathbb{R}^m)$6 and every sequence $W^{1,p}(\Omega;\mathbb{R}^m)$7 such that

$W^{1,p}(\Omega;\mathbb{R}^m)$8

there exist a subsequence, not relabeled, and a new sequence $W^{1,p}(\Omega;\mathbb{R}^m)$9 such that

1. $p\in]1,\infty[$0 in measure on $p\in]1,\infty[$1;
2. $p\in]1,\infty[$2 is equi-integrable on $p\in]1,\infty[$3.

This condition is local in the precise sense that it concerns approximation on the reference cube $p\in]1,\infty[$4 around an affine map $p\in]1,\infty[$5. It replaces an arbitrary bounded-energy sequence by one with the same affine boundary structure, asymptotically the same gradients in measure, and improved integrability of the energy density.

A slightly stronger variant, denoted $p\in]1,\infty[$6, also appears in Section 2 of the paper and is adapted to radially-usc integrands. The principal form, however, is $p\in]1,\infty[$7 (Mandallena, 2011).

2. Relation to lower semicontinuity

The principal lower-semicontinuity theorem using $p\in]1,\infty[$8 concerns the functional

$p\in]1,\infty[$9

Assume:

1. $(C_{p,q})$0 is continuous on $(C_{p,q})$1 and $(C_{p,q})$2-coercive;
2. $(C_{p,q})$3 is $(C_{p,q})$4-quasiconvex, i.e.

$(C_{p,q})$5

where

$(C_{p,q})$6

1. $(C_{p,q})$7 holds.

Under these hypotheses, $(C_{p,q})$8 is sequentially lower-semicontinuous with respect to strong convergence in $(C_{p,q})$9. Equivalently, whenever $L^p$0 in $L^p$1 and $L^p$2,

$L^p$3

The theorem places the localization principle alongside $L^p$4-quasiconvexity as a decisive hypothesis for lower semicontinuity. The role of $L^p$5 is not to replace quasiconvexity, but to provide a local recovery mechanism that permits passage from a general bounded-energy sequence to an equi-integrable one without losing the relevant local gradient information (Mandallena, 2011).

3. Proof strategy and local mechanism

The proof mechanism begins with a sequence $L^p$6 satisfying

$L^p$7

By $L^p$8-coercivity, $L^p$9 is bounded. The fundamental theorem on Young measures then yields a subsequence generating a Young measure $\Omega\subset\mathbb{R}^n$0.

The next step is a blow-up argument around a Lebesgue point of the limit gradient. One sets

$\Omega\subset\mathbb{R}^n$1

and chooses a suitable diagonal sequence $\Omega\subset\mathbb{R}^n$2. This produces a sequence $\Omega\subset\mathbb{R}^n$3 that still generates the same, now homogeneous, Young measure $\Omega\subset\mathbb{R}^n$4 and satisfies

$\Omega\subset\mathbb{R}^n$5

Applying $\Omega\subset\mathbb{R}^n$6 to $\Omega\subset\mathbb{R}^n$7 and $\Omega\subset\mathbb{R}^n$8, one obtains

$\Omega\subset\mathbb{R}^n$9

with

$m,n\ge 1$0

and $m,n\ge 1$1 equi-integrable.

The final step uses continuity of $m,n\ge 1$2 on its closed domain and the weak-Young-measure lower-semicontinuity theorem to conclude

$m,n\ge 1$3

after which the argument is pulled back to the original scale to obtain the pointwise inequality required for the global liminf estimate.

The distinctive feature of this approach is the construction of a local replacement $m,n\ge 1$4 that simultaneously preserves the Young-measure content of the original sequence and yields equi-integrability of $m,n\ge 1$5. In the paper’s presentation, this is the mechanism that converts local information into a usable lower-semicontinuity statement (Mandallena, 2011).

4. Relaxation and integral representation

Once $m,n\ge 1$6 is available, it can be combined with a density hypothesis for piecewise-affine approximation to obtain relaxation theorems. The relaxed functional is defined by

$m,n\ge 1$7

In the closed-domain case, assume:

1. $m,n\ge 1$8, $m,n\ge 1$9;
2. $\mathbb{M}=\mathbb{R}^{m\times n}$0 is closed and $\mathbb{M}=\mathbb{R}^{m\times n}$1 is continuous on $\mathbb{M}=\mathbb{R}^{m\times n}$2;
3. $\mathbb{M}=\mathbb{R}^{m\times n}$3 holds;
4. Sobolev functions can be approximated by piecewise-affine functions in both $\mathbb{M}=\mathbb{R}^{m\times n}$4 and $\mathbb{M}=\mathbb{R}^{m\times n}$5, namely hypothesis $\mathbb{M}=\mathbb{R}^{m\times n}$6.

Then the $\mathbb{M}=\mathbb{R}^{m\times n}$7-relaxation admits the integral representation

$\mathbb{M}=\mathbb{R}^{m\times n}$8

A second relaxation theorem treats the ru-usc or open-domain case. There, continuity is assumed only on $\mathbb{M}=\mathbb{R}^{m\times n}$9, together with additional boundary-behavior conditions $1<p<\infty$0. The resulting representation has the same form, except that $1<p<\infty$1 is replaced by its lower-semicontinuous envelope.

These two theorems show that the localization principle is not confined to lower-semicontinuity questions. It is a central ingredient in deriving full integral representations of relaxed nonconvex energies, provided that the global approximation condition $1<p<\infty$2 is also available (Mandallena, 2011).

5. Exponential-growth integrands and nonconvex relaxation

A main application concerns nonconvex integrands with exponential growth, for example

$1<p<\infty$3

The paper states that, in this setting, one checks that the hypotheses $1<p<\infty$4 and $1<p<\infty$5 are satisfied when $1<p<\infty$6. Consequently, the relaxation of

$1<p<\infty$7

is exactly

$1<p<\infty$8

This application is significant because the integrand $1<p<\infty$9 is allowed to be Borel measurable and not necessarily finite. The localization principle is therefore used beyond the classical polynomial-growth setting. The details emphasize that the framework covers both the standard $1\le q\le\infty$0-growth case and more exotic growth conditions, provided that the relevant technical bounds can be verified.

A plausible implication is that the principle is best understood as a transfer device: it converts local bounded-energy information into the equi-integrable structure needed for both liminf inequalities and relaxation formulas, even when the integrand is nonconvex and may take the value $1\le q\le\infty$1 (Mandallena, 2011).

6. Scope, flexibility, and limitations

The scope of $1\le q\le\infty$2 is deliberately broad. The paper states that it is flexible enough to cover both the classical $1\le q\le\infty$3-growth case and more exotic growth such as exponential growth, once a few technical bounds are checked. At the same time, several limitations are explicit.

First, to pass from lower semicontinuity to a full relaxation theorem one must also approximate arbitrary Sobolev maps by piecewise-affine functions in the $1\le q\le\infty$4-energy, namely condition $1\le q\le\infty$5. The paper identifies this as often the true bottleneck in nonconvex problems.

Second, when $1\le q\le\infty$6 is not finite everywhere, additional radial-usc assumptions and the uniform interior-point conditions $1\le q\le\infty$7 are required so that the boundary of $1\le q\le\infty$8 does not obstruct the integral representation.

Third, the method is perturbative around a fixed exponent $1\le q\le\infty$9. Choosing $W^{1,p}(\Omega;\mathbb{R}^m)$00 often simplifies approximation because of better Sobolev embeddings, but this comes at the cost of requiring a $W^{1,p}(\Omega;\mathbb{R}^m)$01-quasiconvexification.

In summary, the localization principle of Mandallena is a local approximation condition formulated on affine blow-up cells that enables the replacement of arbitrary bounded-energy sequences by equi-integrable competitors in $W^{1,p}(\Omega;\mathbb{R}^m)$02. Within the paper’s framework, that local step is the key bridge from Young-measure analysis to sequential lower semicontinuity and from lower semicontinuity to exact relaxation formulas for a wide class of nonconvex, possibly unbounded integrands (Mandallena, 2011).